Question

Philip has a goal of saving $3000. The function s(t)=500×1.4t
represents the amount of money Philip has saved after t
months.

A. Build a function r(t)
to represent the amount of money Philip has left to save at t
months.
After how many months will Philip have saved at least $3000? Enter your answer as a whole number.

Answer 1

To find the function \( r(t) \) representing the amount of money Philip has left to save at \( t \) months, we can begin by stating that:

- Philip's goal is to save $3000.
- The function \( s(t) = 500 \times 1.4^t \) represents the amount he has saved after \( t \) months.

The amount Philip has left to save, \( r(t) \), can be represented as:

\[
r(t) = 3000 - s(t)
\]

Substituting the function \( s(t) \) into this equation gives us:

\[
r(t) = 3000 - 500 \times 1.4^t
\]

To find out how many months it will take for Philip to save at least $3000, we need to solve for \( t \) when \( s(t) \) is equal to or greater than $3000:

\[
500 \times 1.4^t \geq 3000
\]

Dividing both sides by 500:

\[
1.4^t \geq 6
\]

Now, we take the logarithm of both sides (using natural logarithm, but common logarithm would work just as well):

\[
\log(1.4^t) \geq \log(6)
\]

Using the power rule of logarithms:

\[
t \cdot \log(1.4) \geq \log(6)
\]

Now solve for \( t \):

\[
t \geq \frac{\log(6)}{\log(1.4)}
\]

Calculating the values:

\[
\log(6) \approx 0.7781
\]
\[
\log(1.4) \approx 0.1461
\]

Now, divide these values:

\[
t \geq \frac{0.7781}{0.1461} \approx 5.33
\]

Since \( t \) must be a whole number, we take the next whole number:

\[
t \geq 6
\]

Thus, after **6 months**, Philip will have saved at least $3000.